Admissible currents and first retarded corrections for a fixed-time scalar response

A fixed-time Klein–Gordon response to a linearized vacuum metric, with the scalar held fixed, is reconstructed as a compact tensor distribution with an explicit initial-surface term. On the spacelike-compact vacuum solution space it has a smooth compact conserved representative if and only if its gauge defect vanishes. After a retarded or zero-Cauchy harmonic prescription, the criterion is vanishing of the propagated defect. A compact-Cauchy residual wave proves failure on an open dense family of smearings in a specified Fréchet test space. The same pairing, restricted to any declared Fourier-gap two-dimensional spatial-TT plane, nevertheless has an explicit compact conserved representative, constructed from curvature currents dual to that plane. Separately, the first retarded product of the compactly switched minimal Einstein–scalar vertex with one scalar insertion is computed for an external metric and for a one-oscillator metric. The external coefficient is a linearly smeared free field; its contraction coefficients have scaling degree at most three and unique minimal extensions. The oscillator coefficient reconstructs the switched bilinear and defines an essentially self-adjoint real polynomial on a common core. The Newton–wave composition of stress, inverse, and scalar response is treated as a distinct cubic Wick form. Dense-TT representability, the adiabatic limit, and higher causal products remain open.

A compact current on a 2-plane is not a counterexample to generic failure on $\mathcal{S}$. A linear $R_{1,1}$ is not a cubic Newton–wave form.

The first question is classical. Fix a free scalar solution $\varphi$, a Cauchy surface, and a scalar smearing $f$. The metric-to-scalar response is linear in the metric perturbation. When does that functional equal a pairing with a smooth compact conserved tensor on a declared space of vacuum metric solutions?

The second question is quantum and perturbative. Start from a specified compactly switched vertex, not from a presumed observable. What is its first retarded product with one scalar insertion, and on which domain is the resulting expression represented?

The two questions have different inputs and different conclusions:

A result along one arrow is not a premise completing the other.

Figure fig:object-roadmap separates the two questions and the five objects treated in this manuscript.

Full-space nonrepresentability and finite-plane representability are compatible. The former requires equality on a space containing residual pure gauges. The latter asks for two interpolation conditions on a plane with no nonzero Schwartz pure gauge. Its annihilator is correspondingly much larger.

Likewise, field degree depends on what is quantized. With an external metric, a scalar-quadratic vertex has a linear first retarded scalar coefficient. With one metric oscillator, the coefficient has one oscillator factor and one scalar factor. Substituting a scalar stress into a Newton–wave inverse before forming the classical response is another operation, with a different degree and singularity structure.

We call a smooth compact conserved tensor an FH test tensor, following the observable class of Fewster and Hunt FH. We prove the flat-space representability statements directly. In particular, we do not use radical exhaustion, separation by observables on the full Minkowski quotient, or a compact-Cauchy nondegeneracy theorem.

For the quantum calculation we use an explicitly specified two-insertion Wick normalization. Scaling degree is used in the sense of the extension theory of BF; the required coefficients and their uniqueness are also checked directly. The off-shell input convention and retarded subtraction are those of the retarded-product framework DF. Only one insertion pair is constructed.

The essential self-adjointness conclusion uses the analytic-vector criterion of Nelson Nelson, after explicit number estimates. It does not assert essential self-adjointness of the cubic vertex or existence of an interacting Hamiltonian.

No companion manuscript is a theorem premise. No result about the Enter calculus, a measurement instrument, or named record statistics is needed. There is consequently no Enter-calculus citation.

-goals{ There is no dense-TT representability theorem, adiabatic limit, construction of $T_2(V,V)$, stress–stress product, graviton-loop calculation, instrument, Ramsey record, or contact-separation result. A finite oscillator-valued wave is not treated as a local relativistic metric field.}

Two potential transcription errors are resolved before the proofs.

First, the convention

and our convention $L_h=-L_h^{+}$, $PS=-L_h\varphi$, define the same response, not opposite responses. Their raw currents therefore agree.

Second, with the retarded convention

the represented test function is

An additional minus sign in this formula would contradict both the retarded contraction and the Duhamel equation. Appendix app:signs gives the full audit.

Figure fig:sign-audit summarizes the response and retarded sign conventions fixed throughout the paper.

Work on $M={\mathbb{R}}^{1+3}$ with

The quantum and Newton–wave parts assume $m>0$. The classical classification remains valid for $m=0$, with the sine propagator interpreted continuously at zero.

For a vector field $\xi$, put

Trace reversal is

In four dimensions it is an involution. Direct differentiation gives

The linearized Einstein operator is

Its identities are

They follow by expanding derivatives in (eq:LE).

The scalar-response operator is

For the response with zero Cauchy data at $t_0$,

A smooth field is spacelike compact if its support is contained in $J(C)$ for some compact $C\subset M$. Define

The retarded-prescription domain is

These spaces are not identified with the Schwartz TT plane used later.

The test-tensor space is

A real linear functional $B$ on $\mathcal{V}\subseteq \mathcal{S}$ is FH-representable on $\mathcal{V}$ if a single $J\in \mathcal{E}$ satisfies $B(h)=F_J(h)$ for every $h\in \mathcal{V}$. For the non-spacelike-compact TT plane below, the same compact pairing defines a restricted extension, without identifying that plane with the phase space of an FH algebra.

For every smooth $\xi$ and every $J\in \mathcal{E}$,

This elementary identity is the necessary condition used in every negative result.

The formal adjoints, with covectors paired against vectors, are

For the componentwise massless wave equation on rank-$r$ tensors, write

Thus ${\mathsf{E}}_r$ is advanced minus retarded. If

then the time kernel of ${\mathsf{E}}_r$ is $S_{\mathrm{w}}(t)$.

For the massive scalar, use $G_R,G_A$, and set

The spatial Fourier kernels are

The scalar commutator below uses ${\Delta }_m$, not ${\mathsf{E}}_r$.

Compact distributions are allowed as Green-operator sources, using transposition. In particular,

for compact smooth tests $u$. For smooth spacelike-compact sources bounded below in time, the retarded solution is defined by zero data sufficiently far in the past. All support statements used below follow from finite propagation and these kernels.

We use the following compact-source consequence repeatedly:

In the forward direction the common support lies in $J^{+}(\mathrm{supp}C)\cap J^{-}(\mathrm{supp}C)$. In the reverse direction the Green identities on compact distributions give ${\mathsf{E}}_r\square A=0$.

For this section and the classification, translate the initial surface to $t_0=0$. Fix a real free scalar

with compact smooth Cauchy data, and a real

The scalar is fixed as a classical parameter.

Define

Let

It has zero terminal data and $Pu=-f$. It is smooth, including across a time later than the support of $f$, where it vanishes.

Self-adjointness of the spatial sine operator gives

Put $V=\Theta (t)u$. Its support is contained in the compact set

For every smooth $h$,

The current is a compact distribution of order at most zero. It decomposes as

where $J_{\mathrm{bulk}}$ is (eq:Jraw) with $V=u$, and

Here $u_0=u(0)$. In components,

Proof. Using (eq:L), integrate the derivative of $h$ in $⟨V,L_h\varphi ⟩$ by parts:

The Hessian terms cancel. This proves the current formula first for compact $h$, and then for arbitrary smooth $h$ by inserting a cutoff equal to one near $K_f$.

The derivative of $\Theta u$ is

Substitution gives (eq:surface). Only a bounded bulk density and a slice measure occur. Their support is compact by finite propagation.

Figure fig:raw-current shows the truncated causal support and the initial-surface term in the raw-current reconstruction.

Distributionally,

Consequently

where the covector-valued coefficients are

The density $\gamma$ is smooth and compact in $0<t<T$; $a,b$ are smooth compact spatial data.

Proof. For any scalar distribution $w$, the product rule gives

Use $\square \varphi =m^2\varphi$. Also

This proves (eq:divraw). Apply (eq:Kstar), then use

The resulting coefficients are (eq:Hab).

The symbol $\gamma$ in (eq:H) is a defect density. It is not the compact switch $g$ used later.

For every smooth vector field $\xi$,

If

then

In particular,

Proof. Expanding $P({\xi }^{\nu }{\partial }_{\nu }\varphi )$ gives

The last term is zero and the first two equal $-L_{K\xi }\varphi$, by (eq:DK). Subtracting the homogeneous solution with the same initial data gives the zero-data response. The final formula also follows directly from $K^{\ast }{J}_{\mathrm{raw}}=\mathbf{H}$.

For a fixed $f,\varphi$:

1. annihilation of every $\xi \in {C_c}^{\infty }((0,\infty )\times {\mathbb{R}}^3)$ is equivalent to $\gamma =0$;

2. annihilation of every $\xi \in {C_c}^{\infty }(M)$ is equivalent to $\mathbf{H}=0$, equivalently $\gamma =a=b=0$;

3. annihilation of every smooth spacelike-compact vector wave $\square \xi =0$ is equivalent to ${\mathsf{E}}_1\mathbf{H}=0$.

Proof. The first two assertions follow from (eq:gaugepairing) and the disjoint bulk and slice supports in (eq:H). Arbitrary values and first normal derivatives of a compact test at the slice separate $a$ and $b$.

For necessity in (iii), test with $\xi ={\mathsf{E}}_{1}v$, for arbitrary compact smooth vector tests $v$. The causal propagator is skew-adjoint, so

If ${\mathsf{E}}_1\mathbf{H}=0$, (eq:compactprimitive) gives $\mathbf{H}=\square A$ with $A$ compact. Therefore $⟨\mathbf{H},\xi ⟩=⟨A,\square \xi ⟩=0$ for every residual wave.

Figure fig:three-gauge-tests distinguishes the bulk, complete, and propagated defects tested by the three gauge classes.

For $h\in {\mathcal{D}}_{\mathrm{ret}}$, set

For every spacelike-compact smooth $h$, let $w_0(h)$ solve

and set

Finite propagation gives smooth spacelike-compact compensators. Both prescriptions satisfy $D(\mathfrak{s}h)=0$, and both preserve the vacuum equation.

Define

For compact $\xi$, ${\mathfrak{s}}_{\mathrm{ret}}(K\xi )=0$. For a residual wave $\square \xi =0$, however,

More generally,

where ${\mathsf{H}}_{\mathrm{w}}$ is componentwise massless Cauchy evolution. Thus these are harmonic prescriptions, not sections of the complete gauge quotient.

Set

Then

For a compact metric test $h$,

For $h\in {\mathcal{D}}_{\mathrm{ret}}$, the unambiguous identity instead is

The second pairing has compact support intersection: $D(h)$ vanishes sufficiently far in the past, while $A$ has advanced support from a compact source.

These assertions follow from

and $K^{\ast }{D}^{\ast }A=\square A=\mathbf{H}$. For noncompact $D(h)$, one inserts a compact cutoff on the causal region relevant to the compact source $\mathbf{H}$.

A temporal cutoff makes the boundary issue explicit. Choose $\theta =0$ sufficiently far in the past and $\theta =1$ by the time either $J_{\mathrm{raw}}$ or $D(h)$ can be nonzero. Then

Indeed, $D(\theta h{)}^{\nu }=\theta D(h{)}^{\nu }+({\partial }_{\mu }\theta ){\bar{h}}^{\mu \nu }$. Every displayed pairing has compact support intersection. Conservation of $J_{\sec }$ alone does not authorize deletion of the second term.

Figure fig:harmonic-boundary shows why harmonic prescriptions retain residual gauges and why the boundary pairing must be kept.

Define

The massless kernel gives

It is smooth and spacelike compact, with data

Thus $\zeta =0$ is exactly two compact smooth Cauchy-data equations, not an unspecified global condition.

The advanced potential is

Its jump is $A(0+)-A(0-)=-b$. The resulting Dirac coefficient in $D^{\ast }A$ is

which equals $j_{\Sigma }$, because $b^{\nu }=u_0{\partial }^{\nu }\varphi {|}_0$. The section current has no remaining Dirac measure: it is piecewise smooth and locally bounded.

The following are equivalent:

Proof. The first two are (eq:compactprimitive); compactness of $A$ implies compactness of $J_{\sec }$.

Conversely suppose $J_{\sec }$ is compact. In an early half-space $t<t_{\ast }$, both $J_{\sec }$ and $J_{\mathrm{raw}}$ vanish, so $D^{\ast }A=0$. There $A$ is smooth by (eq:Aexplicit). Trace reversal of (eq:Dstar) shows that $A^{♯}$ satisfies the flat Killing equation. Differentiating that equation and cyclically permuting indices proves that all second derivatives vanish. Hence $A$ is affine in this connected half-space.

Its advanced support confines every spatial slice to a bounded set. An affine field with that property is zero. Thus $A$ vanishes sufficiently far in the past. Its advanced support also bounds it above in time; in that bounded slab, the causal past of its compact source has a uniform spatial bound. Hence $A$ is compact.

A compact conserved distribution is not yet an FH test tensor. We now construct a smooth representative on $\mathcal{S}$.

The causal propagation of $J_{\mathrm{raw}}$ and of $\mathbf{H}$ is smooth. If $J_{\sec }$ is compact, its causal propagation is smooth as well.

Proof. Each current is a finite sum of compact slice distributions with smooth spatial coefficients and piecewise smooth densities on a bounded time interval. Spatial derivatives of those densities have uniform bounds on each time piece, and their spatial supports have a common compact bound.

For a density $c(s)$, causal propagation is

All output-time derivatives can be moved onto the sine propagator. On the bounded source-time interval these propagators act continuously on smooth compact spatial data. Spatial derivatives commute with them. The resulting function is smooth. A source ${\delta }^{(j)}(t-t_{\ast })c_{\ast }(x)$ gives a time derivative of $S_{\mathrm{w}}(t-t_{\ast })c_{\ast }$, also smooth. Finite propagation gives spacelike compactness.

The required decompositions follow from (eq:Jraw), (eq:H), and (eq:Aexplicit); the surface measures cancel in $J_{\sec }$.

Let $C$ be a compact symmetric contravariant tensor distribution with $K^{\ast }C=0$, and suppose

is smooth. Choose a temporal cutoff $\chi$, zero sufficiently far in the past and one sufficiently far in the future. Then

is smooth, compact, and conserved, and

Proof. Commutation of trace reversal and constant-coefficient operators gives

Thus $L_{\mathrm{E}}q=0$ by (eq:LEidentities). The tensor $L_{\mathrm{E}}(\chi q)$ is supported in the transition slab of $\chi$. Since $q$ is spacelike compact, this support is compact. Conservation follows from the Bianchi identity in (eq:LEidentities).

We prove equality first for $h\in {\mathcal{S}}_{\mathrm{dD}}$. Put $d=D(\chi q)$, which is smooth and compact. Taking trace reversal in (eq:LEidentities) gives

The last term pairs to zero with $h$, because $D(h)=0$ and $d$ is compact.

Write $\tilde{C}={\left(C\right)}^{♭♭}$, so $\bar{q}={\mathsf{E}}_2\tilde{C}$, and set $F=\square (\chi \bar{q})$, a smooth compact tensor. Uniqueness of the retarded and advanced Cauchy problems gives

Therefore ${\mathsf{E}}_{2}F=-\bar{q}$ and

By (eq:compactprimitive), $F+\tilde{C}=\square A_2$ with $A_2$ compact. Since $\square h=0$ on ${\mathcal{S}}_{\mathrm{dD}}$,

Consequently $F_{J_C}(h)=-⟨F,h⟩=⟨C,h⟩$.

For general $h\in \mathcal{S}$, the zero-Cauchy compensator gives $h^{\mathrm{dD}}=h-K{w}_0(h)\in {\mathcal{S}}_{\mathrm{dD}}$. Both $C$ and $J_C$ annihilate smooth pure gauges by compact integration by parts. The equality for $h^{\mathrm{dD}}$ therefore implies equality for $h$.

This proof requires neither a symplectic normalization nor an identification of the radical. It is an equality on solutions, not an equality of the original distribution with a smooth tensor off shell.

For fixed $f,\varphi$, the following are equivalent:

When they hold, a representative is

Proof. Representability forces annihilation of every compact pure gauge $K\xi \in \mathcal{S}$, hence $\mathbf{H}=0$ by Proposition prop:threegauges. The decomposition (eq:H) gives the final equivalence. If $\mathbf{H}=0$, the compact current $J_{\mathrm{raw}}$ is conserved; Lemma lem:smoothprop and Theorem thm:smoothing supply the representative.

For fixed $f,\varphi$, the following are equivalent:

1. $B_{\mathrm{raw}}(K\xi )=0$ for every smooth spacelike-compact residual wave $\square \xi =0$;

2. $\zeta ={\mathsf{E}}_1\mathbf{H}=0$;

3. both equations

   $$b={{\int }_0}^T{S}_{\mathrm{w}}(s)\gamma (s)\mathrm{d}s,a=-{{\int }_0}^T{C}_{\mathrm{w}}(s)\gamma (s)\mathrm{d}s$$

   hold;

4. $B_R$ is FH-representable on ${\mathcal{S}}_{\mathrm{ret}}$;

5. $B_0$ is FH-representable on $\mathcal{S}$;

6. $B_{\mathrm{raw}}{|}_{{\mathcal{S}}_{\mathrm{dD}}}$ is FH-representable on ${\mathcal{S}}_{\mathrm{dD}}$.

When they hold, one common representative is

It represents each of the three functionals on its stated domain.

Proof. The equivalence of (i)–(iii) follows from Proposition prop:threegauges and (eq:zetadata). Any of (iv)–(vi) implies (i), because residual pure gauges belong to ${\mathcal{S}}_{\mathrm{dD}}\subset {\mathcal{S}}_{\mathrm{ret}}$, are fixed by both prescriptions, and must be annihilated by compact conserved tests.

Assume (ii). By Lemma lem:compactconditions, $A$ and $J_{\sec }$ are compact. Equation (eq:hybrid) now permits unrestricted compact integration by parts:

For the Cauchy prescription,

so $B_0(h)=⟨J_{\sec },h⟩$ on $\mathcal{S}$. If $h\in {\mathcal{S}}_{\mathrm{dD}}$, the $D^{\ast }A$ term pairs to zero and $⟨J_{\sec },h⟩=B_{\mathrm{raw}}(h)$. Finally apply Lemma lem:smoothprop and Theorem thm:smoothing to $J_{\sec }$.

Figure fig:smoothing-classification summarizes the direct smoothing construction and the distinct raw and harmonic admissibility criteria.

If ${\mathsf{E}}_1\mathbf{H}\ne 0$, none of the three functionals in Theorem thm:harmonic has a smooth compact conserved representative on its stated domain. This excludes all representatives modified by on-shell annihilators, including field-equation terms.

Proof. By Proposition prop:threegauges, a residual wave $\xi$ has $B_{\mathrm{raw}}(K\xi )\ne 0$. An annihilator cannot change that value on a solution. Every smooth compact conserved tensor gives zero by (eq:gaugeannihilation).

For every $m\ge 0$ and $T>0$, there are compact smooth scalar Cauchy data, a nonnegative $f\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3)$, and a nonzero residual pure gauge $K\xi$, with compact smooth vector-wave Cauchy data, such that

Proof. Choose real $F\in {C_c}^{\infty }({\mathbb{R}}^3)$ with ${\partial }_{1}F\not\equiv 0$, and put $\beta ={\partial }_{1}F$. Let

and let the only nonzero component of the vector wave satisfy

Finite propagation gives spacelike compactness, and $(K\xi {)}_{01}{|}_0=\beta$, so $K\xi \ne 0$.

Set $w={\xi }^1{\partial }_1\varphi$. Its value and first time derivative vanish at zero. Equation (eq:boundarygauge) therefore gives $S_0(K\xi )\varphi =w$. Locally uniformly in space,